Objective

Discretize a given freeform surface into panels with the following properties:

1. Panels must be developable

2. Panels should be of approximate equal width

3. Panels should be as straight as possible

4. Panels should bend by their weak axis to aproximate the surface.

Boat-building

The use of sraight developable planks is widely used in traditional boat building

Developable hulls

Connected developable patches for boat hull design.

• Common practice in naval engineering industry.

Frank Ghery

This techniques have also been used in the architecture world, mainly by Frank Ghery.

His façades are usually a collection of connected developable surfaces.

Burj Khalifa

Latest architectural work following this techiques was:

Burj Khalifa by Frank Ghery

Burj Khalifa

It was designed as a collection of:

• Developable surfaces
• Which can be covered by equal width planks
• Surfaces of constant cuvrature
• Which can be covered by repeating the same profile

Burj Khalifa

Geodesic curves

A geodesic curve is the generalization of a straight line into curved spaces.

Geodesic curves

It could be easiy understood as:

The path that a bug follows on a surface if it only moves forward

Straightest geodesics

In this research, we concentrate on the concept of straightest geodesics.

Developable surfaces

Surfaces with 0 gaussian curvature. Meaning, they can be flattened onto a plane without distorsion

Developable surfaces

Developable surfaces:

• can be flattened
• can be generated by a single curve.

and

Geodesic curves:

• are straight lines in a curved space.

Therefore

We wish to generate panels using geodesic curves in order to achieve straight developable panels

Developable surfaces

In other words:

We wish to cover a given freeform surface with a pattern of geodesic curves with equal spacing.

This can only be achieved if the provided surface is already developable.

A compromise exists between the curve spacing and the curve geodesic property

Obtaining Geodesic Patterns

There are three main methods for the obtaining succesfful geodesic patterns:

1. The parallel transport method

2. The evolution method

3. The piecewise geodesic vector-field method

Vector parallel transport

Implementation details

Results

Implementation details

Results

Implementation details

Results

Implementation details

Results

Tangent developable method

Bi-Normal method

Comparisson

Piecewise geodesic vector-fields

Developability of triangle meshes